DOCSLIB.ORG

Operator Methods for Continuous-Time Markov Processes

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Operator Methods for Continuous-Time Markov Processes CHAPTER11 Operator Methods for Continuous-Time Markov Processes Yacine Aït-Sahalia*, Lars Peter Hansen**, and José A. Scheinkman* *Department of Economics, Princeton University,Princeton, NJ **Department of Economics,The University of Chicago, Chicago, IL Contents 1. Introduction 2 2. Alternative Ways to Model a Continuous-Time Markov Process 3 2.1. Transition Functions 3 2.2. Semigroup of Conditional Expectations 4 2.3. Infinitesimal Generators 5 2.4. Quadratic Forms 7 2.5. Stochastic Differential Equations 8 2.6. Extensions 8 3. Parametrizations of the Stationary Distribution: Calibrating the Long Run 11 3.1. Wong’s Polynomial Models 12 3.2. Stationary Distributions 14 3.3. Fitting the Stationary Distribution 15 3.4. Nonparametric Methods for Inferring Drift or Diffusion Coefficients 18 4. Transition Dynamics and Spectral Decomposition 20 4.1. Quadratic Forms and Implied Generators 21 4.2. Principal Components 24 4.3. Applications 30 5. Hermite and Related Expansions of a Transition Density 36 5.1. Exponential Expansion 36 5.2. Hermite Expansion of the Transition Function 37 5.3. Local Expansions of the Log-Transition Function 40 6. Observable Implications and Tests 45 6.1. Local Characterization 45 6.2. Total Positivity and Testing for Jumps 47 6.3. Principal Component Approach 48 6.4. Testing the Specification of Transitions 49 6.5. Testing Markovianity 52 6.6. Testing Symmetry 53 6.7. Random Time Changes 54 © 2010, Elsevier B.V.All rights reserved. 1 2 Yacine Aït-Sahalia et al. 7. The Properties of Parameter Estimators 55 7.1. Maximum Likelihood Estimation 55 7.2. Estimating the Diffusion Coefficient in the Presence of Jumps 57 7.3. Maximum Likelihood Estimation with Random Sampling Times 58 8. Conclusions 61 Acknowledgments 62 References 62 Abstract This chapter surveys relevant tools, based on operator methods, to describe the evolution in time of continuous-time stochastic process, over different time horizons. Applications include modeling the long-run stationary distribution of the process, modeling the short or intermediate run transi- tion dynamics of the process, estimating parametric models via maximum-likelihood, implications of the spectral decomposition of the generator, and various observable implications and tests of the characteristics of the process. Keywords: Markov process; Infinitesimal Generator; Spectral decomposition; Transition density; Maximum-Likelihood; Stationary density; Long-run. 1. INTRODUCTION Our chapter surveys a set of mathematical and statistical tools that are valuable in understanding and characterizing nonlinear Markov processes. Such processes are used extensively as building blocks in economics and finance. In these literatures, typically the local evolution or short-run transition is specified. We concentrate on the continuous limit in which case it is the instantaneous transition that is specified. In understanding the implications of such a modeling approach we show how to infer the intermediate and long-run properties from the short-run dynamics. To accomplish this, we describe operator methods and their use in conjunction with continuous-time stochastic process models. Operator methods begin with a local characterization of the Markov process dynam- ics. This local specification takes the form of an infinitesimal generator. The infinitesimal generator is itself an operator mapping test functions into other functions. From the infinitesimal generator, we construct a family (semigroup) of conditional expectation operators. The operators exploit the time-invariant Markov structure. Each operator in this family is indexed by the forecast horizon, the interval of time between the infor- mation set used for prediction and the object that is being predicted. Operator methods allow us to ascertain global, and in particular, long-run implications from the local or infinitesimal evolution.These global implications are reflected in (a) the implied station- ary distribution, (b) the analysis of the eigenfunctions of the generator that dominate in the long run, and (c) the construction of likelihood expansions and other estimating equations. Operator Methods for Continuous-Time Markov Processes 3 The methods we describe in this chapter are designed to show how global and long-run implications follow from local characterizations of the time series evolution. This con- nection between local and global properties is particularly challenging for nonlinear time series models. Despite this complexity,the Markov structure makes characterizations of the dynamic evolution tractable. In addition to facilitating the study of a given Markov process, operator methods provide characterizations of the observable implications of potentially rich families of such processes. These methods can be incorporated into sta- tistical estimation and testing. Although many Markov processes used in practice are formally misspecificied, operator methods are useful in exploring the specific nature and consequences of this misspecification. Section 2 describes the underlying mathematical methods and notation. Section 3 studies Markov models through their implied stationary distributions. Section 4 develops some operator methods used to characterize transition dynamics including long-run behavior of Markov process. Section 5 provides approximations to transition densities that are designed to support econometric estimation. Section 6 describes the properties of some parameter estimators. Finally, Section 7 investigates alternative ways to characterize the observable implications of various Markov models, and to test those implications. 2. ALTERNATIVE WAYS TO MODEL A CONTINUOUS-TIME MARKOV PROCESS There are several different but essentially equivalent ways to parameterize continuous time Markov processes, each leading naturally to a distinct estimation strategy. In this section, we briefly describe five possible parametrizations. 2.1. Transition Functions In what follows, (, F, Pr) will denote a probability space, S a locally compact metric space with a countable basis, S a σ-field of Borelians in S, I an interval of the real line, and for each t ∈ I, Xt : (, F, Pr) → (S, S) a measurable function. We will refer to (S, S) as the state space and to X as a stochastic process. Definition 1 P : (S × S) →[0, 1) is a transition probability if, for each x ∈ S, P(x, ·) is a probability measure in S, and for each B ∈ S, P(·, B) is measurable. 2 Definition 2 A transition function is a family Ps,t, (s, t) ∈ I , s < t that satisfies for each s < t < u the Chapman–Kolmogorov equation: Ps,u(x, B) = Pt,u(y, B)Ps,t(x,dy). A transition function is time homogeneous if Ps,t = Ps,t whenever t − s = t − s . In this case we write Pt−s instead of Ps,t. 4 Yacine Aït-Sahalia et al. Definition 3 Let Ft ⊂ F be an increasing family of σ-algebras, and X a stochastic process that is adapted to Ft. X is Markov with transition function Ps,t if for each nonnegative Borel measurable φ : S → R and each (s, t) ∈ I 2, s < t, E[φ(Xt)|Fs]= φ(y)Ps,t(Xs,dy). The following standard result (for example,Revuz and Yor,1991;Chapter 3,Theorem 1.5) allows one to parameterize Markov processes using transition functions. Theorem 1 Given a transition function Ps,t on (S, S) and a probability measure Q0 on (S, S), there exists a unique probability measure Pr on S[0,∞), S[0,∞) , such that the coordinate process X is Markov with respect to σ(Xu, u ≤ t), with transition function Ps,t and the distribution of X0 given by Q0. We will interchangeably call transition function the measure Ps,t or its conditional density p (subject to regularity conditions which guarantee its existence): Ps,t(x,dy) = p(y, t|x, s)dy. In the time homogenous case, we write = t − s and p(y|x, ). In the remainder of this chapter, unless explicitly stated, we will treat only the case of time homogeneity. 2.2. Semigroup of Conditional Expectations Let Pt be a homogeneous transition function and L be a vector space of real-valued func- tions such that for each φ ∈ L, φ(y)Pt(x,dy) ∈ L. For each t define the conditional expectation operator Ttφ(x) = φ(y)Pt(x,dy). (2.1) The Chapman–Kolmogorov equation guarantees that the linear operators Tt satisfy: Tt+s = TtTs. (2.2) This suggests another parameterization for Markov processes. Let (L, ·) be a Banach space. Definition 4 A one-parameter family of linear operators in L, {Tt : t ≥ 0} is called a semigroup if (a) T0 = I and (b) Tt+s = TtTs for all s, t ≥ 0.{Tt : t ≥ 0} is a strongly continuous contraction semigroup if, in addition, (c) limt↓0Ttφ = φ, and (d) ||Tt|| ≤ 1. If a semigroup represents conditional expectations, then it must be positive, that is, if φ ≥ 0 then Ttφ ≥ 0. Operator Methods for Continuous-Time Markov Processes 5 Two useful examples of Banach spaces L to use in this context are as follows: Example 1 Let S be a locally compact and separable state space. Let L = C0 be the space of continuous functions φ : S → R, that vanish at infinity.For φ ∈ C0 define: φ∞ = sup |φ(x)|. x∈S A strongly continuous contraction positive semigroup on C0 is called a Feller semigroup. Example 2 Let Q be a measure on a locally compact subset S of Rm. Let L2(Q) be the space of all Borel measurable functions φ : S → R that are square integrable with respect to the measure Q endowed with the norm: 1 2 2 φ2 = φ dQ . In general, the semigroup of conditional expectations determine the finite- dimensional distributions of the Markov process (see e.g. Ethier and Kurtz, 1986; Proposition 1.6 of Chapter 4.) There are also many results (e.g. Revuz and Yor, 1991; Proposition 2.2 of Chapter 3) concerning whether given a contraction semigroup one can construct a homogeneous transition function such that Eq. (2.1) is satisfied. 2.3. Infinitesimal Generators Definition 5 The infinitesimal generator of a semigroup Tt on a Banach space L is the (possibly unbounded) linear operator A defined by: T φ − φ Aφ = lim t .
Load more

Recommended publications
  • Persistent Random Walks. II. Functional Scaling Limits
    Persistent random walks. II. Functional Scaling Limits Peggy Cénac1. Arnaud Le Ny2. Basile de Loynes3. Yoann Offret1 1Institut de Mathématiques de Bourgogne (IMB) - UMR CNRS 5584 Université de Bourgogne Franche-Comté, 21000 Dijon, France 2Laboratoire d’Analyse et de Mathématiques Appliquées (LAMA) - UMR CNRS 8050 Université Paris Est Créteil, 94010 Créteil Cedex, France 3Ensai - Université de Bretagne-Loire, Campus de Ker-Lann, Rue Blaise Pascal, BP 37203, 35172 BRUZ cedex, France Abstract We give a complete and unified description – under some stability assumptions – of the functional scaling limits associated with some persistent random walks for which the recurrent or transient type is studied in [1]. As a result, we highlight a phase transition phenomenon with respect to the memory. It turns out that the limit process is either Markovian or not according to – to put it in a nutshell – the rate of decrease of the distribution tails corresponding to the persistent times. In the memoryless situation, the limits are classical strictly stable Lévy processes of infinite variations. However, we point out that the description of the critical Cauchy case fills some lacuna even in the closely related context of Directionally Reinforced Random Walks (DRRWs) for which it has not been considered yet. Besides, we need to introduced some relevant generalized drift – extended the classical one – in order to study the critical case but also the situation when the limit is no longer Markovian. It appears to be in full generality a drift in mean for the Persistent Random Walk (PRW). The limit processes keeping some memory – given by some variable length Markov chain – of the underlying PRW are called arcsine Lamperti anomalous diffusions due to their marginal distribution which are computed explicitly here.
    [Show full text]
  • Math 353 Lecture Notes Intro to Pdes Eigenfunction Expansions for Ibvps
    Math 353 Lecture Notes Intro to PDEs Eigenfunction expansions for IBVPs J. Wong (Fall 2020) Topics covered • Eigenfunction expansions for PDEs ◦ The procedure for time-dependent problems ◦ Projection, independent evolution of modes 1 The eigenfunction method to solve PDEs We are now ready to demonstrate how to use the components derived thus far to solve the heat equation. First, two examples to illustrate the proces... 1.1 Example 1: no source; Dirichlet BCs The simplest case. We solve ut =uxx; x 2 (0; 1); t > 0 (1.1a) u(0; t) = 0; u(1; t) = 0; (1.1b) u(x; 0) = f(x): (1.1c) The eigenvalues/eigenfunctions are (as calculated in previous sections) 2 2 λn = n π ; φn = sin nπx; n ≥ 1: (1.2) Assuming the solution exists, it can be written in the eigenfunction basis as 1 X u(x; t) = cn(t)φn(x): n=0 Definition (modes) The n-th term of this series is sometimes called the n-th mode or Fourier mode. I'll use the word frequently to describe it (rather than, say, `basis function'). 1 00 Substitute into the PDE (1.1a) and use the fact that −φn = λnφ to obtain 1 X 0 (cn(t) + λncn(t))φn(x) = 0: n=1 By the fact the fφng is a basis, it follows that the coefficient for each mode satisfies the ODE 0 cn(t) + λncn(t) = 0: Solving the ODE gives us a `general' solution to the PDE with its BCs, 1 X −λnt u(x; t) = ane φn(x): n=1 The remaining coefficients are determined by the IC, u(x; 0) = f(x): To match to the solution, we need to also write f(x) in the basis: 1 Z 1 X hf; φni f(x) = f φ (x); f = = 2 f(x) sin nπx dx: (1.3) n n n hφ ; φ i n=1 n n 0 Then from the initial condition, we get u(x; 0) = f(x) 1 1 X X =) cn(0)φn(x) = fnφn(x) n=1 n=1 =) cn(0) = fn for all n ≥ 1: Now everything has been solved - we are done! The solution to the IBVP (1.1) is 1 X −n2π2t u(x; t) = ane sin nπx with an given by (1.5): (1.4) n=1 Alternatively, we could state the solution as follows: The solution is 1 X −λnt u(x; t) = fne φn(x) n=1 with eigenfunctions/values φn; λn given by (1.2) and fn by (1.3).
    [Show full text]
  • Math 356 Lecture Notes Intro to Pdes Eigenfunction Expansions for Ibvps
    MATH 356 LECTURE NOTES INTRO TO PDES EIGENFUNCTION EXPANSIONS FOR IBVPS J. WONG (FALL 2019) Topics covered • Eigenfunction expansions for PDEs ◦ The procedure for time-dependent problems ◦ Projection, independent evolution of modes ◦ Separation of variables • The wave equation ◦ Standard solution, standing waves ◦ Example: tuning the vibrating string (resonance) • Interpreting solutions (what does the series mean?) ◦ Relevance of Fourier convergence theorem ◦ Smoothing for the heat equation ◦ Non-smoothing for the wave equation 3 1. The eigenfunction method to solve PDEs We are now ready to demonstrate how to use the components derived thus far to solve the heat equation. The procedure here for the heat equation will extend nicely to a variety of other problems. For now, consider an initial boundary value problem of the form ut = −Lu + h(x; t); x 2 (a; b); t > 0 hom. BCs at a and b (1.1) u(x; 0) = f(x) We seek a solution in terms of the eigenfunction basis X u(x; t) = cn(t)φn(x) n by finding simple ODEs to solve for the coefficients cn(t): This form of the solution is called an eigenfunction expansion for u (or `eigenfunction series') and each term cnφn(x) is a mode (or `Fourier mode' or `eigenmode'). Part 1: find the eigenfunction basis. The first step is to compute the basis. The eigenfunctions we need are the solutions to the eigenvalue problem Lφ = λφ, φ(x) satisfies the BCs for u: (1.2) By the theorem in ??, there is a sequence of eigenfunctions fφng with eigenvalues fλng that form an orthogonal basis for L2[a; b] (i.e.
    [Show full text]
  • Lectures on Lévy Processes, Stochastic Calculus and Financial
    Lectures on L¶evyProcesses, Stochastic Calculus and Financial Applications, Ovronnaz September 2005 David Applebaum Probability and Statistics Department, University of She±eld, Hicks Building, Houns¯eld Road, She±eld, England, S3 7RH e-mail: D.Applebaum@she±eld.ac.uk Introduction A L¶evyprocess is essentially a stochastic process with stationary and in- dependent increments. The basic theory was developed, principally by Paul L¶evyin the 1930s. In the past 15 years there has been a renaissance of interest and a plethora of books, articles and conferences. Why ? There are both theoretical and practical reasons. Theoretical ² There are many interesting examples - Brownian motion, simple and compound Poisson processes, ®-stable processes, subordinated processes, ¯nancial processes, relativistic process, Riemann zeta process . ² L¶evyprocesses are simplest generic class of process which have (a.s.) continuous paths interspersed with random jumps of arbitrary size oc- curring at random times. ² L¶evyprocesses comprise a natural subclass of semimartingales and of Markov processes of Feller type. ² Noise. L¶evyprocesses are a good model of \noise" in random dynamical systems. 1 Input + Noise = Output Attempts to describe this di®erentially leads to stochastic calculus.A large class of Markov processes can be built as solutions of stochastic di®erential equations driven by L¶evynoise. L¶evydriven stochastic partial di®erential equations are beginning to be studied with some intensity. ² Robust structure. Most applications utilise L¶evyprocesses taking val- ues in Euclidean space but this can be replaced by a Hilbert space, a Banach space (these are important for spdes), a locally compact group, a manifold. Quantised versions are non-commutative L¶evyprocesses on quantum groups.
    [Show full text]
  • A Stochastic Processes and Martingales
    A Stochastic Processes and Martingales A.1 Stochastic Processes Let I be either IINorIR+.Astochastic process on I with state space E is a family of E-valued random variables X = {Xt : t ∈ I}. We only consider examples where E is a Polish space. Suppose for the moment that I =IR+. A stochastic process is called cadlag if its paths t → Xt are right-continuous (a.s.) and its left limits exist at all points. In this book we assume that every stochastic process is cadlag. We say a process is continuous if its paths are continuous. The above conditions are meant to hold with probability 1 and not to hold pathwise. A.2 Filtration and Stopping Times The information available at time t is expressed by a σ-subalgebra Ft ⊂F.An {F ∈ } increasing family of σ-algebras t : t I is called a filtration.IfI =IR+, F F F we call a filtration right-continuous if t+ := s>t s = t. If not stated otherwise, we assume that all filtrations in this book are right-continuous. In many books it is also assumed that the filtration is complete, i.e., F0 contains all IIP-null sets. We do not assume this here because we want to be able to change the measure in Chapter 4. Because the changed measure and IIP will be singular, it would not be possible to extend the new measure to the whole σ-algebra F. A stochastic process X is called Ft-adapted if Xt is Ft-measurable for all t. If it is clear which filtration is used, we just call the process adapted.The {F X } natural filtration t is the smallest right-continuous filtration such that X is adapted.
    [Show full text]
  • Lectures on Multiparameter Processes
    Lecture Notes on Multiparameter Processes: Ecole Polytechnique Fed´ erale´ de Lausanne, Switzerland Davar Khoshnevisan Department of Mathematics University of Utah Salt Lake City, UT 84112–0090 [email protected] http://www.math.utah.edu/˜davar April–June 2001 ii Contents Preface vi 1 Examples from Markov chains 1 2 Examples from Percolation on Trees and Brownian Motion 7 3ProvingLevy’s´ Theorem and Introducing Martingales 13 4 Preliminaries on Ortho-Martingales 19 5 Ortho-Martingales and Intersections of Walks and Brownian Motion 25 6 Intersections of Brownian Motion, Multiparameter Martingales 35 7 Capacity, Energy and Dimension 43 8 Frostman’s Theorem, Hausdorff Dimension and Brownian Motion 49 9 Potential Theory of Brownian Motion and Stable Processes 55 10 Brownian Sheet and Kahane’s Problem 65 Bibliography 71 iii iv Preface These are the notes for a one-semester course based on ten lectures given at the Ecole Polytechnique Fed´ erale´ de Lausanne, April–June 2001. My goal has been to illustrate, in some detail, some of the salient features of the theory of multiparameter processes and in particular, Cairoli’s theory of multiparameter mar- tingales. In order to get to the heart of the matter, and develop a kind of intuition at the same time, I have chosen the simplest topics of random walks, Brownian motions, etc. to highlight the methods. The full theory can be found in Multi-Parameter Processes: An Introduction to Random Fields (henceforth, referred to as MPP) which is to be published by Springer-Verlag, although these lectures also contain material not covered in the mentioned book.
    [Show full text]
  • Introduction to Lévy Processes
    Introduction to L´evyprocesses Graduate lecture 22 January 2004 Matthias Winkel Departmental lecturer (Institute of Actuaries and Aon lecturer in Statistics) 1. Random walks and continuous-time limits 2. Examples 3. Classification and construction of L´evy processes 4. Examples 5. Poisson point processes and simulation 1 1. Random walks and continuous-time limits 4 Definition 1 Let Yk, k ≥ 1, be i.i.d. Then n X 0 Sn = Yk, n ∈ N, k=1 is called a random walk. -4 0 8 16 Random walks have stationary and independent increments Yk = Sk − Sk−1, k ≥ 1. Stationarity means the Yk have identical distribution. Definition 2 A right-continuous process Xt, t ∈ R+, with stationary independent increments is called L´evy process. 2 Page 1 What are Sn, n ≥ 0, and Xt, t ≥ 0? Stochastic processes; mathematical objects, well-defined, with many nice properties that can be studied. If you don’t like this, think of a model for a stock price evolving with time. There are also many other applications. If you worry about negative values, think of log’s of prices. What does Definition 2 mean? Increments , = 1 , are independent and Xtk − Xtk−1 k , . , n , = 1 for all 0 = . Xtk − Xtk−1 ∼ Xtk−tk−1 k , . , n t0 < . < tn Right-continuity refers to the sample paths (realisations). 3 Can we obtain L´evyprocesses from random walks? What happens e.g. if we let the time unit tend to zero, i.e. take a more and more remote look at our random walk? If we focus at a fixed time, 1 say, and speed up the process so as to make n steps per time unit, we know what happens, the answer is given by the Central Limit Theorem: 2 Theorem 1 (Lindeberg-L´evy) If σ = V ar(Y1) < ∞, then Sn − (Sn) √E → Z ∼ N(0, σ2) in distribution, as n → ∞.
    [Show full text]
  • 6 Mar 2019 Strict Local Martingales and the Khasminskii Test for Explosions
    Strict Local Martingales and the Khasminskii Test for Explosions Aditi Dandapani∗† and Philip Protter‡§ March 7, 2019 Abstract We exhibit sufficient conditions such that components of a multidimen- sional SDE giving rise to a local martingale M are strict local martingales or martingales. We assume that the equations have diffusion coefficients of the form σ(Mt, vt), with vt being a stochastic volatility term. We dedicate this paper to the memory of Larry Shepp 1 Introduction In 2007 P.L. Lions and M. Musiela (2007) gave a sufficient condition for when a unique weak solution of a one dimensional stochastic volatility stochastic differen- tial equation is a strict local martingale. Lions and Musiela also gave a sufficient condition for it to be a martingale. Similar results were obtained by L. Andersen and V. Piterbarg (2007). The techniques depend at one key stage on Feller’s test for explosions, and as such are uniquely applicable to the one dimensional case. Earlier, in 2002, F. Delbaen and H. Shirakawa (2002) gave a seminal result giving arXiv:1903.02383v1 [q-fin.MF] 6 Mar 2019 necessary and sufficient conditions for a solution of a one dimensional stochastic differential equation (without stochastic volatility) to be a strict local martingale or not. This was later extended by A. Mijatovic and M. Urusov (2012) and then by ∗Applied Mathematics Department, Columbia University, New York, NY 10027; email: [email protected]; Currently at Ecole Polytechnique, Palaiseau, France. †Supported in part by NSF grant DMS-1308483 ‡Statistics Department, Columbia University, New York, NY 10027; email: [email protected].
    [Show full text]
  • Sturm-Liouville Expansions of the Delta
    Gauge Institute Journal H. Vic Dannon Sturm-Liouville Expansions of the Delta Function H. Vic Dannon [email protected] May, 2014 Abstract We expand the Delta Function in Series, and Integrals of Sturm-Liouville Eigen-functions. Keywords: Sturm-Liouville Expansions, Infinitesimal, Infinite- Hyper-Real, Hyper-Real, infinite Hyper-real, Infinitesimal Calculus, Delta Function, Fourier Series, Laguerre Polynomials, Legendre Functions, Bessel Functions, Delta Function, 2000 Mathematics Subject Classification 26E35; 26E30; 26E15; 26E20; 26A06; 26A12; 03E10; 03E55; 03E17; 03H15; 46S20; 97I40; 97I30. 1 Gauge Institute Journal H. Vic Dannon Contents 0. Eigen-functions Expansion of the Delta Function 1. Hyper-real line. 2. Hyper-real Function 3. Integral of a Hyper-real Function 4. Delta Function 5. Convergent Series 6. Hyper-real Sturm-Liouville Problem 7. Delta Expansion in Non-Normalized Eigen-Functions 8. Fourier-Sine Expansion of Delta Associated with yx"( )+=l yx ( ) 0 & ab==0 9. Fourier-Cosine Expansion of Delta Associated with 1 yx"( )+=l yx ( ) 0 & ab==2 p 10. Fourier-Sine Expansion of Delta Associated with yx"( )+=l yx ( ) 0 & a = 0 11. Fourier-Bessel Expansion of Delta Associated with n2 -1 ux"( )+- (l 4 ) ux ( ) = 0 & ab==0 x 2 12. Delta Expansion in Orthonormal Eigen-functions 13. Fourier-Hermit Expansion of Delta Associated with ux"( )+- (l xux2 ) ( ) = 0 for any real x 14. Fourier-Legendre Expansion of Delta Associated with 2 Gauge Institute Journal H. Vic Dannon 11 11 uu"(ql )++ [ ] ( q ) = 0 -<<pq p 4 cos2 q , 22 15. Fourier-Legendre Expansion of Delta Associated with 112 11 ymy"(ql )++- [ ( ) ] (q ) = 0 -<<pq p 4 cos2 q , 22 16.
    [Show full text]
  • Lecture 15.Pdf
    Lecture #15 Eigenfunctions of hermitian operators If the spectrum is discrete , then the eigenfunctions are in Hilbert space and correspond to physically realizable states. It the spectrum is continuous , the eigenfunctions are not normalizable, and they do not correspond to physically realizable states (but their linear combinations may be normalizable). Discrete spectra Normalizable eigenfunctions of a hermitian operator have the following properties: (1) Their eigenvalues are real. (2) Eigenfunctions that belong to different eigenvalues are orthogonal. Finite-dimension vector space: The eigenfunctions of an observable operator are complete , i.e. any function in Hilbert space can be expressed as their linear combination. Continuous spectra Example: Find the eigenvalues and eigenfunctions of the momentum operator . This solution is not square -inferable and operator p has no eigenfunctions in Hilbert space. However, if we only consider real eigenvalues, we can define sort of orthonormality: Lecture 15 Page 1 L15.P2 since the Fourier transform of Dirac delta function is Proof: Plancherel's theorem Now, Then, which looks very similar to orthonormality. We can call such equation Dirac orthonormality. These functions are complete in a sense that any square integrable function can be written in a form and c(p) is obtained by Fourier's trick. Summary for continuous spectra: eigenfunctions with real eigenvalues are Dirac orthonormalizable and complete. Lecture 15 Page 2 L15.P3 Generalized statistical interpretation: If your measure observable Q on a particle in a state you will get one of the eigenvalues of the hermitian operator Q. If the spectrum of Q is discrete, the probability of getting the eigenvalue associated with orthonormalized eigenfunction is It the spectrum is continuous, with real eigenvalues q(z) and associated Dirac-orthonormalized eigenfunctions , the probability of getting a result in the range dz is The wave function "collapses" to the corresponding eigenstate upon measurement.
    [Show full text]
  • Part C Lévy Processes and Finance
    Part C Levy´ Processes and Finance Matthias Winkel1 University of Oxford HT 2007 1Departmental lecturer (Institute of Actuaries and Aon Lecturer in Statistics) at the Department of Statistics, University of Oxford MS3 Levy´ Processes and Finance Matthias Winkel – 16 lectures HT 2007 Prerequisites Part A Probability is a prerequisite. BS3a/OBS3a Applied Probability or B10 Martin- gales and Financial Mathematics would be useful, but are by no means essential; some material from these courses will be reviewed without proof. Aims L´evy processes form a central class of stochastic processes, contain both Brownian motion and the Poisson process, and are prototypes of Markov processes and semimartingales. Like Brownian motion, they are used in a multitude of applications ranging from biology and physics to insurance and finance. Like the Poisson process, they allow to model abrupt moves by jumps, which is an important feature for many applications. In the last ten years L´evy processes have seen a hugely increased attention as is reflected on the academic side by a number of excellent graduate texts and on the industrial side realising that they provide versatile stochastic models of financial markets. This continues to stimulate further research in both theoretical and applied directions. This course will give a solid introduction to some of the theory of L´evy processes as needed for financial and other applications. Synopsis Review of (compound) Poisson processes, Brownian motion (informal), Markov property. Connection with random walks, [Donsker’s theorem], Poisson limit theorem. Spatial Poisson processes, construction of L´evy processes. Special cases of increasing L´evy processes (subordinators) and processes with only positive jumps.
    [Show full text]
  • Geometric Brownian Motion with Affine Drift and Its Time-Integral
    Geometric Brownian motion with affine drift and its time-integral ⋆ a b, c Runhuan Feng , Pingping Jiang ∗, Hans Volkmer aDepartment of Mathematics, University of Illinois at Urbana-Champaign, Illinois, USA bSchool of Management and Economics,The Chinese University of Hong Kong, Shenzhen, Shenzhen, Guangdong, 518172, P.R. China School of Management, University of Science and Technology of China, Hefei, Anhui, 230026, P.R.China cDepartment of Mathematical Sciences, University of Wisconsin–Milwaukee, Wisconsin, USA Abstract The joint distribution of a geometric Brownian motion and its time-integral was derived in a seminal paper by Yor (1992) using Lamperti’s transformation, leading to explicit solutions in terms of modified Bessel functions. In this paper, we revisit this classic result using the simple Laplace transform approach in connection to the Heun differential equation. We extend the methodology to the geometric Brownian motion with affine drift and show that the joint distribution of this process and its time-integral can be determined by a doubly-confluent Heun equation. Furthermore, the joint Laplace transform of the process and its time-integral is derived from the asymptotics of the solutions. In addition, we provide an application by using the results for the asymptotics of the double-confluent Heun equation in pricing Asian options. Numerical results show the accuracy and efficiency of this new method. Keywords: Doubly-confluent Heun equation, geometric Brownian motion with affine drift, Lamperti’s transformation, asymptotics, boundary value problem. ⋆R. Feng is supported by an endowment from the State Farm Companies Foundation. P. Jiang arXiv:2012.09661v1 [q-fin.MF] 17 Dec 2020 is supported by the Postdoctoral Science Foundation of China (No.2020M671853) and the National Natural Science Foundation of China (No.
    [Show full text]